Expert Answers to Algebra 2 Problems

Write all the points where $\frac{x^2-<!--\; -\; -->18x+56}{x^3-<!--\; -\; -->3x^2-<!--\; -\; -->22x+24}$ is discontinuous.

Calculate all the points of discontinuity of $\frac{v^2-<!--\; -\; -->17v+42}{(v-<!--\; -\; -->2)(v^2-<!--\; -\; -->12v+36)}$.

What is the limit of difference between harmonic series and natural logarithm of n+1?
I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series ($A_n=1+\frac{1}{2}+...+\frac{1}{n}$) and this natural logarithm $L=\ln <!--\; \; -->(1+n)$ and asks me to see what happens with the difference $A_n-<!--\; -\; -->L$ when $n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$. I've already done some symbolic computations in Matlab and the answer I get is "eulergamma". So, after some online research I found out that eulergamma is the Euler-Mascheroni constant (being 0.5772....). Not knowing too much about this, I answered the question as follows: "I notice that $\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}(A_n-<!--\; -\; -->L)$ behave same as $\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}(A_n-<!--\; -\; -->\ln <!--\; \; -->(n))$ which is equal to Euler-Mascheroni constant. So, the requested difference $A_n-<!--\; -\; -->L$ is equal to Euler-Mascheroni constant as $n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$."
Teacher asked me "Why does this $A_n-<!--\; -\; -->L$ behave in the same way as this $A_n-<!--\; -\; -->\ln <!--\; \; -->(n)$? Try to do something. Add/subtract $\ln <!--\; \; -->(n)$ for example."
Afterall here I am and asking for your help. I tried to add/subtract $\ln <!--\; \; -->(n)$ but nothing came up. Could you please help me with this, or at least give me some clues/hints? I don't have such an experience on this!!
Thank you for your attention!

convexity of log of moment generating function
Why is log of a moment generating function of random variable Z is convex? that is
$\log <!--\; \; -->\mathbb{E}[\exp <!--\; \; -->(\mathrm{\lambda <!--\; \lambda \; -->}.Z)]$
My logic says since expectation is linear so it is in particular convex and exponential is convex therefore $\mathbb{E}[\exp <!--\; \; -->(\mathrm{\lambda <!--\; \lambda \; -->}.Z)]$ is convex but how to know if apllying log doesnt affect convexity?

In the case of rational function, when taking the first derivative we take the numerator and find for which point the function is equal to zero.
Now, for the second derivative, can we take the second derivative just of the numerator? it is not intuitive, as we need to take the second derivative of the whole function, but I did not find a counterexample
For example the function: $x+8+\frac{50}{2x-<!--\; -\; -->4}$

How do you solve this logarithm?
Solve for $x$ in the following:
$x=9^{{\log }_3<!--\; \; -->\left(2\right)}$
The answer is $4$, but why?

Mean Value theorem problem?(inequality)
I'm trying to solve this mean value theorem problem but confused where to start,
If $0<a<b$ prove that $(1-<!--\; -\; -->\frac{a}{b})<\ln <!--\; \; -->\frac{b}{a}<\frac{b}{a}-<!--\; -\; -->1$
Can someone please lend me a hand?

This seems very obvious and I am having a bit of trouble producing a formal proof.
sketch proof that the composition of two polynomials is a polynomial
Let
$$\begin{gathered} p(z_1)=a_n{z}_1^n+a_{n-<!--\; -\; -->1}{z}_1^{n-<!--\; -\; -->1}+...+a_1{z}_1+a_0 \\ q(z_2)=b_n{z}_2^n+b_{n-<!--\; -\; -->1}{z}_2^{n-<!--\; -\; -->1}+...+b_1{z}_2+b_0 \end{gathered}$$
be two complex polynomials of degree $n$ where $a_n,..,a_0\in <!--\; \in \; -->\mathbb{C}$ and $b_n,..,b_o\in <!--\; \in \; -->\mathbb{C}$.
Now,
$\begin{array}{rl}(p\circ <!--\; \circ \; -->q)(z_2)& =p(q(z_2))\text{(by definition)}\\ & =a_n(q(z_2){)}^n+a_{n-<!--\; -\; -->1}(q(z_2){)}^{n-<!--\; -\; -->1}+...+a_1(q(z_2))+a_0\end{array}$
which is clearly a complex polynomial of degree $n^2$.
sketch proof that the composition of two rational functions is a rational function
A rational function is a quotient of polynomials.
Let
$a(z_1)=\frac{p(z_1)}{q(z_1)},b(z_2)=\frac{p(z_2)}{q(z_2)}$
Now,
$\begin{array}{rl}(a\circ <!--\; \circ \; -->b)(z_2)& =a(b(z_2))\text{(by definition)}\\ & =\frac{p\left(\frac{p(z_2)}{q(z_2)}\right)}{q\left(\frac{p(z_2)}{q(z_2)}\right)}\\ & =\frac{a_n{\left(\frac{p(z_2)}{q(z_2)}\right)}^n+a_{n-<!--\; -\; -->1}{\left(\frac{p(z_2)}{q(z_2)}\right)}^{n-<!--\; -\; -->1}+...+a_1\left(\frac{p(z_2)}{q(z_2)}\right)+a_0}{b_n{\left(\frac{p(z_2)}{q(z_2)}\right)}^n+b_{n-<!--\; -\; -->1}{\left(\frac{p(z_2)}{q(z_2)}\right)}^{n-<!--\; -\; -->1}+...+b_1\left(\frac{p(z_2)}{q(z_2)}\right)+b_0}\end{array}$
Notice that ${\left(\frac{p(z_2)}{q(z_2)}\right)}^i(i=n,n-<!--\; -\; -->1,..,0)$ is a polynomial as
$(f\circ <!--\; \circ \; -->g)(z_2)=f(g(z_2))={\left(\frac{p(z_2)}{q(z_2)}\right)}^i$
where
$f(x)=x^i,g(z_2)=\left(\frac{p(z_2)}{q(z_2)}\right)$
are both polynomials. Hence $(a\circ <!--\; \circ \; -->b)(z_2)$ is a rational function as it is the quotient of polynomials.

Write all the points where $\frac{u^2-<!--\; -\; -->27u+182}{u^3+3u^2-<!--\; -\; -->9u+5}$.

Find all the points of discontinuity of the function $\frac{w^2-<!--\; -\; -->22w+117}{(w^2-<!--\; -\; -->1)(w^3-<!--\; -\; -->9w^2+23w-<!--\; -\; -->15)}$.

Write all the points where $\frac{y^2-<!--\; -\; -->10y+16}{(y^2-<!--\; -\; -->3y-<!--\; -\; -->4)(y^3+y^2-<!--\; -\; -->32y-<!--\; -\; -->60)}$ is discontinuous.

What are the points of discontinuity of the function $\frac{a^2-<!--\; -\; -->23a+126}{(a+5)(a^3+a^2-<!--\; -\; -->37a+35)}$?

Write all the points where $\frac{m^2-<!--\; -\; -->16m+60}{(m-<!--\; -\; -->4)(m^3-<!--\; -\; -->7m^2-<!--\; -\; -->25m+175)}$ is discontinuous.

Evaluating: ${\int <!--\; \int \; -->}_1^n[\ln <!--\; \; -->(x)-<!--\; -\; -->\ln <!--\; \; -->(\lfloor <!--\; \lfloor \; -->x\rfloor <!--\; \rfloor \; -->)]dx$
I am attempting to evaluate the integral:
${\int <!--\; \int \; -->}_1^n\ln <!--\; \; -->(x)-<!--\; -\; -->\ln <!--\; \; -->(\lfloor <!--\; \lfloor \; -->x\rfloor <!--\; \rfloor \; -->)dx$
To a form:
$f(x)+O(g(x))$
where $g(x)\to <!--\; \backslash rightarrow\; -->0$ as $x\to <!--\; \backslash rightarrow\; -->\mathrm{\infty <!--\; \infty \; -->}$
How do I compute that f(x) or atleast some type of series representation for it?

I have a rational function:
$y=\frac{x^2-<!--\; -\; -->3x}{x^2+2x-<!--\; -\; -->48}$
The explanation I was given to find the x-intercepts was: "let y = 0, and solve for x. Basically you just set the numerator of the fraction equal to 0 and factor."
What does "basically" mean? Is this always the case? Do you simply set the numerator expression equal to zero, and always ignore the contents of the denominator? If so, why do you ignore the contents of the denominator?

Problem with re-arranging a solution of mine
So this is the question:
$h(x)=4\exp <!--\; \; -->(x-<!--\; -\; -->4)h^{-<!--\; -\; -->1}(x)=\ln <!--\; \; -->\left({\displaystyle \frac{\boxed{{\displaystyle \phantom{X}}}}{\boxed{{\displaystyle \phantom{X}}}}}\right)\boxed{{\displaystyle \phantom{XXX}}}$
It wants me to enter in the inverse function of the log on the left side of the photo. I have my answer, $(e^y+16)/4$, first of all, I have no idea if it is right or not, secondly, How would I transform it to fit in the areas given in the field provided on the right side? And if my answer is wrong, please do tell my where my mistakes are, there is one fraction, and one box that can be anything, as in $+x$ or $-<!--\; -\; -->y$, thank you in advance

Find all the points of discontinuity of the function $\frac{y^2-<!--\; -\; -->14y+40}{y^5+4y^4-<!--\; -\; -->39y^3-<!--\; -\; -->86y^2+280y}$.

Calculate all the points of discontinuity of $\frac{s^2-<!--\; -\; -->17s+52}{s^3-<!--\; -\; -->3s^2-<!--\; -\; -->18s}$.

Write all the points where $\frac{u^2-<!--\; -\; -->24u+135}{(u^2+3u-<!--\; -\; -->18)(u^3-<!--\; -\; -->6u^2-<!--\; -\; -->u+30)}$ is discontinuous.